The computation of the Betti and Hodge numbers follows the strategy in [1]} for the \(OG10\)  case and is summarized below. We consider two special manifolds, both admitting Lagrangian fibrations onto \(\mathbb {P}^3\) . The first, denoted by \(\widetilde{M}\) , is of \(OG6\)  type and the second, denoted by \(N\) , is deformation equivalent to a generalized Kummer variety of dimension six. We note that the Lagrangian fibration \(\widetilde{M}\rightarrow \mathbb {P}^3\)  is the same as the ones considered in [2]}, [3]}. The two fibrations \(\widetilde{M}\rightarrow \mathbb {P}^3\)  and \(N \rightarrow \mathbb {P}^3\)  are linked via a group scheme \(G \rightarrow \mathbb {P}^3\)  which acts fiberwise on both \(\widetilde{M}\)  and \(N\) . In particular, \(\widetilde{M}\)  contains an open dense subvariety which is a \(G\) -torsor over the entire base and \(N\)  contains an open dense subvariety which is a \(G\) -torsor over an open dense subvariety strictly contained in the base. Given this relationship between \(\widetilde{M}\)  and \(N\) , we are able to use the known cohomology of generalized Kummer varieties, as well as some smaller dimensional fibrations, to deduce the Betti and Hodge numbers of \(\widetilde{M}\)  by working in an appropriate Grothendieck group. Noting that the pure Hodge structure of \(N\)  and these smaller dimensional fibrations can all be described in terms of the Hodge structure of an Abelian surface \(J\) , we then describe the Hodge structure of \(\widetilde{M}\)  in terms of the Hodge structure of \(J\)  (see Theorem REF ). As in the \(OG10\)  case, the main technical tool that allows us to compare the cohomology of \(\widetilde{M}\)  and \(N\)  is the Ngô Support Theorem (see Section REF ). A key difference with the \(OG10\)  case is that the fibers of the group scheme \(G\)  become disconnected over various loci in the base.
